Equation (11-1) can be formally integrated to give the outside area required to transfer the total heat load QT:
o Jq u0 aT
To integrate Eq. (11-3), Uo and AT must be known as functions of Q. For some problems, Uo varies strongly and nonlinearly throughout the exchanger. In these cases, it is necessary to evaluate Uo and A T at several intermediate values and numerically or graphically integrate. For many practical cases, it is possible to calculate a constant mean overall coefficient Uom from Eq. (11-2) and define a corresponding mean value of A Tm, such that
Care must be taken that Uo does not vary too strongly, that the proper equations and conditions are chosen for calculating the individual coefficients, and that the mean temperature difference is the correct one for the specified exchanger configuration.
Mean Temperature Difference The temperature difference between the two fluids in the heat exchanger will, in general, vary from point to point. The mean temperature difference (A Tm or MTD) can be calculated from the terminal temperatures of the two streams if the following assumptions are valid:
1. All elements of a given fluid stream have the same thermal history in passing through the exchanger.*
2. The exchanger operates at steady state.
3. The specific heat is constant for each stream (or if either stream undergoes an isothermal phase transition).
4. The overall heat-transfer coefficient is constant.
Countercurrent or Cocurrent Flow If the flow of the streams is either completely countercurrent or completely cocurrent or if one or both streams are isothermal (condensing or vaporizing a pure component with negligible pressure change), the correct mTd is the logarithmic-mean temperature difference (LMTD), defined as
for countercurrent flow (Fig. 11-1a) and
(t;- t'i) - (t2 -1'2) LMTD = A Tlm =-—----(11-5b)
If U is not constant but a linear function of AT, the correct value of
* This assumption is vital but is usually omitted or less satisfactorily stated as "each stream is well mixed at each point." In a heat exchanger with substantial bypassing of the heat-transfer surface, e.g., a typical baffled shell-and-tube exchanger, this condition is not satisfied. However, the error is in some degree offset if the same MTD formulation used in reducing experimental heat-transfer data to obtain the basic correlation is used in applying the correlation to design a heat exchanger. The compensation is not in general exact, and insight and judgment are required in the use of the MTD formulations. Particularly, in the design of an exchanger with a very close temperature approach, bypassing may result in an exchanger that is inefficient and even thermodynamically incapable of meeting specified outlet temperatures.
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